sorenson chpt 16

8/13/2019 Sorenson Chpt 16

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Physics inNuclear MedicineSecond Edition

James A. Sorenson, Ph.D.Director, Medical PhysicsProfessor of RadiologyDepat1meht of RadiologyUniversity of Utah Medical CenterSalt Lake City, UtahMichael E. Phelps, Ph.D.Jennifer Jones Simon ProfessorChief, Division of Nuclear Medicineand BiophysicsDepat1ment of Radiological SciencesUCLA School of MedicineChief, Laboratory of Nuclear MedicineLaboratory of Biomedical andEnvironmental Sciences, UCLALos Angeles, California

WeB. SAUNDERS COMPANYHarcourt Brace Jovanovich, Inc.Philadelphia London Toronto Montreal Sydney Tokyo

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16The Anger Camera:Performance CharacteristicsB. DESIGN AND PERFORMANCE CHARACTERISTICSOF PARALLEL-HOLE COLLIMATORS

1. Basic Limitations in Collimator PerformanceThe collimator is a "weak link" for the performance of an Anger

camera system, as indeed it is in any nuclear medicine imaging systememploying the principles of absorptive collimation. Collimator efficiency,defined as the fraction of'Y rays striking the collimator that actually passthrough it to project the 'Y-ray image onto the detector, is typically only afew percent or less. Collimator resolution, which refers to the sharpnessor detail of the 'Y-ray image projected onto the detector, also is ratherpoor, generally worse than the intrinsic resolution of the camera detectorand electronics.

Because it is a limiting factor in camera system performance, it isimportant that the collimator be designed carefully. Poor design can resultonly in poorer overall performance. Design considerations for parallel-hole collimators will be discussed in this section. Design characteristicsfor converging and diverging collimators are similar to those of theparallel-hole type. Design characteristics of pinhole collimators will notbe discussed in detail here but are described in references listed at the endof th~ chapter. The analysis to be presented for parallel-hole collimators


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332 Physics in Nuclear Medicineis similar to that presented by Anger in reference 1, which may beconsulted or a more detaileddiscussion.

2. Septal ThicknessA primary consideration n collimator design s to ensure hat septalpenetration by "Y ays crossing rom one collimator hole into another snegligibly small. This is essential f an accurate "Y-ray mage is to beprojected by the collimator onto the cameradetector. No thickness ofseptal material s sufficient o stop all "Y ays, so the usual criteria is toaccept some reasonably small level of septal penetration, e.g., -5percent.The required septal thickness may be determined by analysis ofFigure 16-11.The shortestpath length or "Y ays to travel from one holeto the next is w. Septal hickness is related o w, and to the length I anddiameterd of the collimator holes, byt = 2dw/(1 w) (16-1)

If septalpenetration s to be less than 5 percent, he transmission actorfor the thicknessw must be

e-IJ.W:$ 0.05 (16-2)where IJ. s the linear attenuation coefficient of the septal material. Sincee-3 - 0.05, his mplies

IJ.w 3 (16-3)w > 3/IJ. (16-4). ..,.::.:-.:,: .:: :.;.',:... . . .

LLIMATORSEPTA

I-d-l I-tFig. 16-11. Minim~m path length w for a 'Y ray passingthrough the collimator septa from one hole to the nextdepends on length I and diameter d of the collimator holesand on septal hicknesses.


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The Anger Camera:PerformanceCharacteristics 333and thus

6d/fJ.t ~ (16-5)I - (3/fJ.)

It is desirable hat septal hickness be as small as possible so that thecollimator septa obstruct the smallestpossiblearea of detector surfaceand collimator efficiency s maximized.This objective s realizedby usinga material with a large value of fJ. or the collimator septa. Materials ofhigh atomic number Z and high density p are preferred. Lead (Z = 82,p = 11.34 g/cm3) s the material of choice for reasons of cost andavailability; however, other materials, ncluding tantalum (Z = 73, p =16.6g/cm3), ungsten Z = 74, p = 19.4g/cm3),and even gold (Z = 79,p = 19.3g/cm3)have been employed n experimentalapplications.Attenuation coefficientsof heavy elementsdependstrongly on 'V-rayenergy n the nuclear medicine energy range (Chapter9, Section B.l).Thus the required septal thickness also dependsstrongly on the 'V-rayenergy for which the collimator is designed o be used. Commerciallyavailable collimators are categorizedaccording o the maximum V-rayenergy for which their septal thickness s considered o be adequate.Low-energycollimators generallyhave an upper limit of about 150keVarid medium-energy ollimators about 400keV. High-energycollimators(e.g., 1 MeV) are not availablecommerciallyexcept by specialorder.

Example 16-1.Calculate he septal thickness equired for low-energy (150 keY)and medium-energy400keY) lead collimators having hole diam-eters 0.25 cm and lengths2.5 cm.Answer.The linear attenuation coefficient of lead at 150 keY is fJ./= 1.89cm2/gx 11.34 /cm3= 21.43 cm-l and at 400 keY is fJ./= 0.22 cm2/gx 11.34g/cm3= 2.49 cm-l (Appendix D). Therefore rom Equa-tion 16-5 or the low-energycollimator

6 x 0.25/21.43t~ 32.5-- 21.43~ 0.030cm ","

and for the medium-energy ollimator


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334 Physics in Nuclear Medicine6 X 0.25/2.49t~ 32.5-- 2.49

~ 0.465 cm

Thickness needed for low-energy collimators are only a few tenths ofa millimeter, which is in the range of lead' 'foil" thicknesses andapproaches the limits of lead thicknesses that can be used without loss ofnecessary mechanical strength. Indeed, low-energy collimators generallyare quite fragile, and their septa can be damaged easily by mechanicalabuse (dropping, stacking on sharp objects, etc.). Medium-energy colli-mators require substantially greater thicknesses, typically a few millime-ters of lead.Low-energy 'Y-ray emiters (e.g., 99mTc,140keY) can be imaged usingmedium-energy collimators. This is done, however, with an unnecessarysacrifice of collimator efficiency because the collimator septa are unnec-essarily thick. (See Table 16-2 for comparative efficiencies of low- andmedium-energy collimators.) Low-energy collimators are used wheneverpossible to obtain maximum collimator efficiency. When choosing acollimator, however, one must consider not only the energy of the 'Yraysto be imaged but also the energies of any other 'Y rays emitted by theradionuclide of interest or by other radionuclides that may be present aswell, e.g., residual activity from another study or radionuclide impurities.Higher-energy 'Y rays may be recorded by Compton downscatter into alower-energy analyzer window. If the collimator septa are too thin, thecollimator may be virtually transparent to higher-energy 'Y ays, causing arelatively intense "foggy" background image to be superimposed on thedesired image, with a resulting loss of image contrast. Whether or not alow-energy collimator can be used when higher-energy 'Y ays are pr~sentdepends on the energy and intensity of those emissions and requiresexperimental evaluation in specific cases.

3. Geometry of Collimator HolesCollimator performance also is affected by the geometry of thecollimator holes, specifically, their shape, length, and diameter.The preferred hole shape, to maximize the exposed area of detectorsurface for a given septal thickness, is round or hexagonal, with the holesarranged in a close-packed hexagonal array. Square and triangular holes

also have been used.


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The Anger Camera: Performance Characteristics 335Collimator hole length and diameter affect strongly both collimatorresolution and collimator efficiency. Collimator resolution Rcis defined asthe full width at half-maximum (FWHM) of the radiation profile from a

point or line source of radiation projected by the collimator onto thedetector (Figure 16-12). This profile is also called the point or line spreadfunction (PSF or LSF). Collimator resolution Rc is given bytRc = d(le + b)/le (16-6)

where b is the distance from the radiation source to the collimator, and dis the diameterand Ie= 1- 2~-1 he "effective ength"of the collimatorholes (Figure 16-12). Here ~ is the linear attenuation coefficient of thecollimator material. Th'e effective length of the collimator holes issomewhat less than their actual length due to septal penetration. FromExample 16-1 t can be seen that for low-energy collimators (150 keY), thedifference between effective and actual length is about 0.1 cm whereas formedium-energy collimators it is about 0.8 cm.

t It should be noted that some versions of Equation 16-6 nclude additional correctionterms involving the thickness of the detector crystal, reflecting the fact that the imageactually is formed at some depth within the detector crystal. Because photons of differentenergies penetrate to different average depths within the crystal, the correction actually isphoton-energy dependent, a point not noted by most authors. The correction is small and forsimplicity is omitted from Equation 16-6, as well as from Equations 16-10 and 16-13 or theconverging and diverging collimators presented later in this chapter.

COLLIMATOR RESOLUTION,-,/ ,PROJECTED ,// \\RADIATION -I \ -FWHMPROFI LE : \\I \I ,I' , --- 'II 1\) 11111111.~1111 tl

MULTIHOLE :--COLLIMATOR \\ ; d\ I\ I

\ I b I\ ~'/\ II IPOINT OR IILINE SOURCE

Fig. 16-12. Radiationprofile (point or line spread unction,PSF or LSF) for a parallel-hole ollimator.The FWHM (fullwidth at half-maximum) f the profile s used o characterizecollimator resolution.


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336 Physics n NuclearMedicineExample16-2.Calculate the resolution (FWHM) of the low-energy collimatordescribed n Example 16-1,at sourcedepthsb = 0 and b = 10cm,

assumingt has a septal hicknessof 0.03 cm.Answer.The effective ength of the collimator is

Ie = 2.5 cm - (2/21.43) m= 2.4 cmThus, for b = 0

Rc = 0.25(2.4+ 0)/2.4cm= 0.25 cm

and at b = 10 cmRc = 0.25(2.4+ 10)/2.4 m

= 1.3 cmThis example illustrates the strong dependenceof collimatorresolution on sourcedistance rom the collimator.Collimator efficiency g, defined as the fraction of 'Y rays passingthrough the collimator per 'Y ay emitted by the source, s given by

g = ~(d/le)2[J2/(d+ t)2] (16-7)where is septal hicknessand K is a constant hat dependson hole shape(~0.24 for round holes n a hexagonal rray, ~0.26 for hexagonal oles na hexagonalarray, ~0.28 for squareholes n a squarearray)Equation 16-7applies o a source n air and assumes o attenuationof radiation by interveningbody tissues.Severalaspectsof Equations 16-6and 16-7should be noted. First,resolution mprovesas the ratio of hole diameter o effective ength (d/l;)is made smaller. Long, narrow holes provide- mage with the bestresolution;however,collimator efficiencydecreases pproximatelyas hesquareof the ratio of hole diameter o length, (d/I;)2.Thus an approximaterelationshipbetweencollimator resolutionRc and efficiencyg is

g ocR~ (16-8)


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The Anger Camera:PerformanceCharacteristics 337Example 16-3.Calculate the efficiency g of the collimator described in Examples16-1 and 16-2, assuming it has hexagonal holes in an hexagonal

array.Answer.For hexagonal holes in a hexagonal array, K = 0.26

g = (0.26)2 (0.25/2.4)2 (0.25)2 (0.25 + 0.03)2]= (0.0676) x (0.0109)x (0.797)= 5.85 x 10-4 (photons transmitted/photons emitted)

This example llustrates the relatively small fraction of emitted Yrays that are transmittedby a typical Anger cameracollimator.Therefore,or a given septal thickness,collimator resolution s improvedonly at the expenseof decreasedcollimator efficiency, and vice versa.The implications of this important tradeoff in collimator design arediscussed urther in Chapter 18, SectionC.2.Equation 16-7 also demonstrates he effect of septal thickness onefficiency. As noted in Section B.l, medium-energy ollimators havelower efficiencies han low-energy collimators becauseof their greaterseptal hicknesses.lQ addition to providing low- and medium-energy ollimators, man-ufacturersof Anger camerasystemsalso provide a selectionof collima-tors with different combinationsof resolution and efficiency. Those withgood resolution but poor efficiency generally are described as "highresolution" collimators, whereas hose with the oppositecharacteristicsare describedas "high sensitivity" collimators. Those with characteris-tics intermediate o the extremesare referred to as "general purpose,""all purpose," or by other similar names.Equation 16-6 ndicates hat collimator resolutionbecomes oorer assource-to-collimator istanceb increases.Thus structuresclosest o thecollimator are imagedwith sharpestdetail. Figure 16-13shows graphi-cally the'relationship betweencollimator resolution and source-to-colli-mator distance or threedifferentcollimatorsprovidedby one commercialmanufacturer.Typically, collimator resolutiondeterioratesby a factor of2 at a distanceof 4-5 cm from the collimator.On the other hand, according o Equation 16-7,collimator efficiencyfor a source n air is independent f source-to-collimator istanceb. Thisrather surprising result is obtained provided the counting rate for the

~


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338 Physics in Nuclear Medicine.entire detector area is measured.The reason for this is illustrated byFigure 16-14.As the source s moved arther away from the collimator,the efficiency with which radiation is transmitted through anyonecollimator hole decreasesn proportion to l/b2 (inverse-squareaw), butthe number of holes through which radiation can pass to reach thedetector ncreasesn proportion o b2.The two effectscanceleachother,with the result that total counting rate--and thus collimator efficiency-does not changewith source-to-collimator istance.Another illus,trationof this effect is shown n Figure 16-15.As source-to-collimator istanceincreases, he maximum height of the PSF or LSF decreases,but thewidth increases and resolution becomespoorer), so that the total areaunder the curve (total detectorcountingrate) does not change.

lnvarianceof collimator efficiencywith source-to-collimator,distanceapplies to point sources, ine sources,and uniform sheet sources n airwith parallel-holecollimators; however, t appliesonly to uniform sheetsourceswith converging,diverging, or pinhole collimators (Section C).When the source s embedded t different depths n the patient, attenua-tion effects also must be considered.Septal penetration and scatter ofphotons rom the walls of the collimator holes also are not considered nthe above analysis.2.

~ I.~ ,.~LJ... I.z~ I.t-;:)6 I.cnlJJQ: O.Q:0 0.6t-


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The Anger Camera: Performance Characteristics 339

SINGLE-HOLEEFFICIENCY l/b2DETECTOR i ~(1 RADIATIONEXAp~ESAED ~~-~~~--~---~~-~---/\ SOURCE b 2 ---~---~-

..- b ~

\ 'COll I MA TORDETECTORFig. 16-14. Explanation for constant counting rate (collimator efficiency) versus

source-to-collimator distance for a point source in air and a parallel-hole collima-tor. Efficiency for a single hole decreases as l/b2, but number of holes passingradiation (area of detector exposed) increases as b2.

Table 16-2summarizeshe performance haracteristicsof a numberof collimators provided by one commercial manufacturer. Collimatorresolution is the FWHM for a source at 10 cm from the face of thecollimator. Collimator efficiency g refers to the relative number of'Y raystransmitted by the collimator per 'Y ray emitted by the source. Note thatthe approximate relationship between collimator efficiency and resolution

COUNTING RATE

CE-TO-COLlIMATORANCE

DISTANCEFig. 16-15. Point spread functions versus distance for a parallel-hole collimator.Area under curve is proportional to collimator efficiency and does not change withdistance.

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340 Physics in Nuclear Medicinegiven by Equation 16-8 s verified by thesedata. Note also the relativelysmall valuesof collimator efficiency.

4. System ResolutionThe sharpness of images recorded with an Anger camera is limited byseveral factors, including intrinsic resolution, collimator resolution, scat-tered radiation, and septal penetration. In terms of the FWHM of a pointor line spread function, the most important factors are the intrinsicresolution Rj of the detector and electronics and the collimator resolutionRc. The combined effect of these two factors is to produce a systemresolution Rs that is somewhat worse than either one alone. Systemresolution Rs (FWHM) is given by ,Rs = YR1 + R~ (16-9)

Because collimator resolution depends on source-to-collimator dis-tance, system resolution also depends on this parameter. Figure 16-16shows system resolution versus source-to-collimator distance for a typi-cal parallel-hole collimator and different values of intrinsic resolution. Ata distance of 5-10 cm (typical depth of organs inside the body), systemresolution is much poorer than intrinsic resolution and is determinedprimarily by collimator resolution. There are significant differencesbetween system resolutions for cameras having substantially differentintrinsic resolutions (e.g., 4 mm versus 8 mm), but the difference insystem resolutions for cameras having small differences in intrinsicresolutions (e.g., 4 mm versus 5 mm) is minor and not clinicallyTable 16-2Performance Characteristics of Some TypicalCommercially Manufactured Parallel-Hole Collimators

Suggested Resolution RMaximum (FWHM at 10Collimator Type Energy (keV) Efficiency g cm)Low energy, 150 1.84 x 10-4 7.4 mmhigh resolutionLow energy, 150 2.68 x 10-4 9.1 mm

general purposeLow energy, 150 5.74 x 10~4 . 13.2 mmhigh sensitivityedium energy, 400 1.72 x 10-4 c 13.4 mm

high sensitivityAdapted rom ref. 3: Hine GJ, Erickson JJ: Advances n scintigraphicnstruments,nHine GJ, Sorenson A (eds): Instrumentation n Nuclear Medicine (vol 2). New York,'~"" . ~... '. "


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The Anger Camera: Performance Characteristics 341

2.0E 1.8u~ 1.6:I:3:I-. 1.4z0 1.2I-3 1.00(J)w 0.8II:~ 0.6 TYPICALI- ORGAN~ 0.4 DEPTHS(J) O. ::::::::i::::::::::;::::::;::lli:: :::~:~:::

0 Ii 2 4 6 8 10 12 14 16SOURCE-TO-COLLIMATOR ISTANCE cm)

Fig. 16-16. System esolution versussource-to-collimatordistance or a typical parallel-holecollimator and for dif-ferent values of intrinsic resolution. At most typical organdepths, system esolution s determinedprimarily by colli-mator resolution.

significant. Small differences in intrinsic resolution may be apparent onbar pattern images or on images of very superficial structures in thepatient, but they usually are not apparent on images of deeper-lyingstructures.System resolution also is degraded by scattered radiation. This isdiscussed in Chapter 18, Section B. The method for combining compo-nent resolutions to determine system resolution also is discussed inAppendix G.

C. PERFORMANCE HARACTERISTICS FCONVERGING,DIVERGING,AND PINHOLECOLLIMATORSEquations or collimator resolution, Rc, and efficiency, g, for con-verging, divergingand pinhole collimators are as follows:

~


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342 Physics n NuclearMedicineCONVERGING COLLIMATOR

Rc = [d(l~ + b)/1~][1/cos9][1 (1~/2)/(f+ l~)] (16-10).g = K2(d/l~)2[d2/(d + t)2][f2/(f - b)2] (16-11)

l~ = (I - 2~-1)/cos9 (16-12)DIVERGING COLLIMATOR

Rc = [d(l~ + b)/1~][I/cos9][ 1+(1~f2j)] (16-13)g = K2(d/l'e)2[d2/(d+)2][f+ l)/(f+ I + b)] (16-14)

PINHOLECOLLIMATORRc = de(l + b)/l (16-15)

g = de cos39/16b2 (16-16)de = \/d[d + 2~-ltan(a/2)] (16-17)

The parametersn theseequationsare as shown n Figure 16-17.Forthe pinhole collimator, de s the" effective" pinhole diameter,accountingfor penetration of the edgesof the pinhole aperture. The equations orcollimator resolution Rc refer to the equivalentFWHM of the point- or

~ f1\ f. ' '~I-d

T I I TaT..( I ~ ..( W I~ .{+ I \ : i 10 IT , tf -it 1 W l iw l1 V'8 ~, Ie\,CONVERGING DIVERGING PINHOLE

Fig. 16-17. Parameters for collimator resolution, R (FWHM), and efficiency, g(transmitted/emitted photons), for Equations (16-17). Adapted with permissionfrom Medical Physics Data Book, Padika1TN, Fivozinsky SP (eds). WashingtonDC, National Bureau of Standards, 1982, pp. 48-55.


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The Anger Camera: Performance Characteristics 34320 >- 250 B

uz- ...~16 ~ 200- ...z ...0 ...-= /2 ~ 1503 NG ~0 .../ C/) ~~ 8 @ 100~ '"... ...f- >~ 4 ;:: 50 PINHC/) ~ I...~2 05 10 15 20 5 10 15 20

SOURCE-TO-COLLIMATOR DISTANCE(cm)Fig. 16-18. Penonnance characteristics (left, system resolution; right, point-sourcegeometricefficiency n air) versus source-to-collimator istance or fourdifferent ypes of Anger camera ollimators. From Rollo FD, Harris CC: Factorsaffecting mage onnation, in Rollo FD (ed): Nuclear Medicine Physics, nstru-mentation,and Agents. St.Louis, C.V. Mosby Co., 1977.Modified from MoyerRA: J Nucl. Med. 15:59,1974].line-source spread function, corrected for magnification or minification ofthe image by the collimator (Equations 15-3, 15-5, and 15-6). Thus, if thecollimator projects a profile with a 2 cm FWHM measured on the detectorand the image magnification factor is x2, the equivalent FWHM in theimaged plane is 1 cm.The equations above may be compared with Equations 16-6 and 16-7for parallel-hole collimator. They are similar except for the presence ofadditional terms involving collimator focal lengths f and, for off-axissources, the angle 8 between the source, the focal point (or pinhole), andthe central axis of the collimator (Figure 16-7). The equations illustratethat for converging and diverging collimators, resolution is best at thecenter (8 = 0, cos 8 = 1).

The performance characteristics of different types of collimators arecompared in Figure 16-18 which shows system resolution and efficiencyvs. distance, including effects of camera intrinsic resolution as well ascollimator magnification. Figure 16-18 llustrates that resolution always isbest with'the source as close as possible to the collimator. Changes incollimator efficiency with diStance depend on whether the radiationsource is a point source or a uniform sheet source. For a point source(Figure 16-18, right), collimator efficiency increases with increasingsource-to-collimator distance with the converging collimator. Maximumefficiency is obtained at the collimator convergence point (~35 cm),where 'Y ays are transmitted through all of the collimator holes, and then

..-


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344 Physics in Nuclear Medicinedecreasesbeyond that point. Point-source collimator efficiency decreaseswith distance for the diverging and pinhole collimators, more severely forthe latter. For an extended, large-area sheet source, sufficiently large to .cover the entire field of view of the collimator, efficiency does not changewith source-to-collimator distance for all of these collimators. Again, forsources embedded within a patient, attenuation effects also must beaccounted for. "Figure 16-18 llustrates that the converging collimator offers the bestcombination of resolution and efficiency at typical imaging distances (5-10cm); however, the field of view is also somewhat limited at 'thesedistances (Equation 15-6, Example 15-2), and for this reason convergingcollimators are most useful with cameras having relatively large areadetectors. Diverging collimators offer a larger imaging area (Example15-1) but at the cost of both resolution and efficiency. Pinhole collimatorsoffer very good resolution and reasonable efficiency at close distances butlose efficiency very rapidly with distance; they also have a quite limitedfield of view because of magnification effects at typical imaging distances(Equation 15-3). Generally they are used for imaging smaller organs thatcan be positioned close to the collimator, e.g., thyroid and heart. Theyalso are useful with high-energy "{-ray emitters because they do not sufferfrom septal penetration problems.Differences between the resolution and field of view obtained atdifferent source-to-collimator distances with parallel-hole, converging,diverging, and pinhole collimators are further illustrated by Figure 16-19.

ATCOLLIMATOR jjACE ~}

DI~XNCE ...

20cm...ISTANCE ,,;

cP VERG NcG PARALLEL HOLE CONVEcRGING .'::rFig. 16-19. Bar pattern magesdemonstrating hanging ield sizeand resolutionobtainedvs. distance or three collimator ypes.


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The Anger Camera:PerformanceCharacteristics 345It should also be noted that the distortions caused by changing magnifi-cation with depth for different structures inside the body sometimes makeimages obtained with the converging, diverging, and pinhole collimatorsdifficult to interpret (see Figure 15-8).

REFERENCES1. Anger HO: Radioisotopecameras, n Hine GJ (ed): Instrumentation inNuclear Medicine (vol 1). New York, AcademicPress, 1967,chap 192. Strand S-E, Larsson I: Image artifacts at high photon ftuence rates in

single-crystalNaI(Tl) scintillation cameras. Nucl Med 19:407-413, 9783. Hine GJ, Erickson JJ: Advances n scintigraphic nstruments, n Hine GJ,Sorenson A (eds): nstrumentation n Nuclear Medicine (vol 2). New York,AcademicPress, 1974,chap 14. Hine GJ, Paras D, Warr CP: Recent advances n gamma-cameramaging.Proc SPIE 152:121-126, 978?Additional discussions of Anger camera performance characteristics andcollimator design are discussed in the following:Hine GJ, Paras P, Warr CP: Measurement of the Performance Parameters ofGamma Cameras, Part I. Rockville, MD., U.S. Dept. H.E.W., Pub .. No.(FDA)78-8049, 1977Richardson RL: Anger scintillation camera, in Rollo FD (ed): Nuclear MedicinePhysics, Instrumentation, and Agents. St. Louis, C. V. Mosby Co., 1977,chap 6

sorenson chpt 16

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